Arbitrage possibilities in Russian spot and future markets based on Revised final report on the EERC grant 98-0571
V.M.Chetverikov Introduction
A stock market of any country is a very difficult object for assumptions and studies. The main reason is that prices in the stock market, in comparison to the market of material goods, are mainly defined by expectations of the participants in the trading process than by external objective economical factors. The internal factor is especially strong in the formation of developing markets, where it is most obvious that the stock and financial markets are relatively independent.
In the classical works it is assumed that there is a firm relationship between the stock prices and the expected discounted flow of dividends. This approach is not acceptable for the estimation of the Russian stock market, since the companies prefer to pay no dividends or the dividends are miserable. Thus in Russia there is no stable correlation between the price of a stock share and financial perspectives of its emitter.
The speculative interests of the market participants and any information of possible merges of companies play the crucial role.
The Russian futures stock market has been actively functioning almost during two years till May 1998, when the main trading place --- the Russian stock market --- ceased to operate. This time is characterized by a rapid, almost five-times, increase in prices on the stock market and by an even more rapid decrease in them. How did the futures market react on this dynamics? To what extent could the futures price of contracts forecast the future spot price of stocks? The present paper tries to give answers to these questions.
In Section 1, to analyze the situation in the stock market in the period from September 1995 to September 1998, we divide the entire observation time into intervals each of which is characterized by it own political and economical situation in the country.
Due to the existing tradition, all the prices are in American dollars but the trade is carried out in Russian roubles. The exchange course is defined by the Central Bank of Russia. There were few main considerations that we used in the description of the dynamics of the Russian stock market. If for the most part the stock trade is carried out by Russian residents without any significant participation of foreign capital, then one hardly can expect any significant diversion in the dynamics of stock prices and that of the gold and currency reserves of the Central Bank. In the Russian market there is a high correlation between the RTS Index and the gold and currency reserves of the Central Bank. If, in addition, there is no direct correlation between changes in the reserves and the net foreign trade deficit, one can assume that the most important role in the Russian stock market is played by intentions of foreign investors and possible changes in the political situation are more important for them than for the national investors.
In this period, the short-term market of state bonds (GKO) and related debt instruments was a very important reference point in the financial market. To describe this very complicated market, we study the following three indices: GKO-90, GKO-180,
GKO-270, which are prices of some synthetic bonds with definite payment periods.
Thus the time series describing the dynamics of synthetic bond indices, the RTS index, and the gold and currency reserves of the Central Bank form the basis for the descriptive analysis of the stock market
Looking for analogs of the Russian stock market among the stock markets of other countries, we paid our attention to the stock market in Venezuela whose dynamics is very similar to that in Russia.
In Section 2 of the present paper we study some theoretical aspects related to realization of simple arbitrage strategies. As far as we know, many of these strategies were successfully used by some broker companies in the Russian futures and spot markets. Here we also formulate some criteria, which allow one to decide, taking into account the GKO market and the borrowed capital market, whether a situation arising in the stock markets is favorable for using a particular arbitrage strategy.
In Section 3 we statistically analyze possible arbitrage strategies in the Russian stock market during the period from November 1996 to May 1998 by using, as an example, the two most liquid stocks of NK Lukoil and RAO EES. This analysis shows that there were rather long time periods during which pure arbitrage operations were possible, i.e., it was possible to «make money from air». The existence of this possibility made the Russian financial market very attractive, and one could hardly expect that banks would invest in the real sector of economy where the rate of return was not very large.
Moreover, there were rather long time periods when quasi-arbitrage operations were possible. The rate of return obtained in these operations by participants possessing a primary capital was significantly larger than that obtained in the GKO-market under the same costs.
Despite these facts, the futures market could provide average estimates of relative changes in the spot prices of stocks without any statistically significant deviations.
Thus the existence of a large number of possibilities for realization of arbitrage strategies does not contradict the fact that there are unbiased estimates of the rate of return via known prices of futures contracts.
1. History of the stock and internal State debt markets in Russia in 1995-98 . The actual market history of the Russian economy originates from 1992--1994.
Several fundamental laws in privatization and credit-money policy adopted in this period laid the foundation of the existing Russian market institutes.
It should be noted that, in practice, although a law is adopted, this does not mean that all existing financial and economical institutes and economy objects transform immediately. The process of constructing new market institutes is rather long. Only starting from 1995, one can say that the Russian market institutes and the market infrastructure operate in the same way as it is known in developed western countries.
Since each separate market simultaneously deals with numerous purchase-and- sell objects (stocks, bonds, futures contracts, etc.), one can completely describe the stock and GKO markets only by introducing some aggregate instruments, i.e., stock market indices.
There are many methods for constructing stock market indices, which differ in the base access (in quantity and the name of stocks included into the calculational set of the index) and in the calculational formulas. In many countries, several tens of different indices are sometimes calculated in the stock markets. These indices show, for instance, the dynamics of company stocks in different industry branches or of stocks comparable in liquidity (the daily trade volume). How to analyze the behavior of different stock market indices is the theme of a separate paper. Here we restrict ourselves to the study of the RTS index of the stock market and to three indices of the GKO--OFZ market.
The RTS index. The RTS index is being calculated from September 1, 1995. The RTS listing includes stocks of practically all the largest Russian companies (except for RAO "Gasprom"). The RTS index is weighted with respect to the company capitals and thus shows their market value. The index access referred to September 1, 1998 includes 88 stocks. The RTS index can be calculated by the recurrent formula
where Cap is the capitalization of the stock market at a corresponding time instant. At the initial time instant (September 1, 1995) the index was equal to 100.
During the four years under study, the RTS index is de-facto the most important factor of the entire Russian stock market for the majority of participants in the market. It is related to the large trade volume in the RTS (much more than the corresponding factor of the stock markets) and to a large amount of bonds listed in the RTS (this factor of the RTS is also larger than that of the stock markets).
In contrast to any unweighted (speculative) index, the RTS index is much more stable under sharp oscillations of prices of single stocks, since the main contribution to the index is made by companies with large capitalization. As a rule, the stocks of such companies are extremely tolerant to a momentary behavior of individual players.
To demonstrate an essential difference in the dynamics of the Russian and American stock markets, in Fig. 1 in Appendix we show the RTS index and the Dow- Jones average during the time period in question. The solid line shows the polynomial trends of these indices. A rapid increase and a more rapid decrease in prices in the stock market are special features of the state of affairs in the Russian stock market from September 1995 to October 1998.
The GKO-OFZ market indices. In contrast to the stock market, where the stock market indices are widely used, aggregate indices are less common in the state bond market. First of all, this is related to the fact that it is impossible to formulate a fixed access for such an index, since all GKO issues have a bounded lifetime. At the same time, the effect of studying several issues separately is very small, since they, first,
1 1*
−
= −
t t t
t Cap
I Cap I
strongly depend on the behavior of the separate operators and, second, have a clear dynamics during their life, i.e., the price is increasing while the interest is decreasing.
Thus the main distinction between the «usual» stock market indices and the GKO indices is that the latter must have a «floating» access. In other words, the set of issues contained in the base access must vary constantly. For each separate issue, its payment period (in days) is a criterion for including it into the access (or for excluding from it).
Since the most important factors characterizing the bonds (the price and the interest) depend on the payment period, it is not a lucky idea to include all bonds into the same index. One of possible solutions of this problem is to use several GKO-indices (for example, three), which characterize the short-term, medium-term, and long-term bonds, respectively. This distinction also has a historical background. During 1993—1996 there was a strict division of issues (three month bonds 21XXX, half-year bonds 22XXX, year bonds 23XXX). Thus we propose to use indices that, at each time instant, include bonds shown in Table1.
Table 1.
Index Number of days till the
repayment of bonds included into the index
Base period of maturity in days (T)
short-term index GKO_90 60-120 90
medium-term index GKO_180
150-210 180
long-term index GKO_270 240-300 270
To calculate the corresponding price index and the daily trade volume of all bonds included into the index, we calculate the price of a synthetic bond by the formula
) (
* ) (
* ) ( ) 1
(
1 1
t v t Yield t
v t
Yield
n
i
i n i
i i
T
∑
∑
= ==
T t Yield t
P
T
T ( ) *
365
* ) 100
( =
Here T= 90, 180, or 270 for GKO_90, GKO_180, and GKO_270, respectively;
n is the number of GKO issues traded on the corresponding day t and having the date of maturity in the required interval (T-30; T+30),
P(t) is the price of the corresponding synthetic bond on the day t, Vi(t) is the trade volume hf the i-th issue on the day t,
365 is the number of days in the year, 100 is the (nominal) price of the bond.
The total volume of all traded bonds in a given interval is the trade volume for the desired synthetic bond.
Figure 1 shows how the four aggregate indices of the Russian stock market depend on time (compared during the same time period). The following time series are considered: the RTS index (the RTS variable), the gold and currency reserves of the Central Bank of the Russian Federation (the GCR_4 variable) in mill. Doll. USA, and the prices of GKPR_180 and GKPR_90 of the synthetic GKO with 180- and 90-day periods of payment. The scale of the variables GKPR_180 and GKPR_90 corresponds to the nominal price 100. We used the data observed from February 1, 1996, to August 14, 1998, i.e., to the time instant at which the Government defaulted on payments for short- term bonds. Since GKO-bonds were sold four days a week, all our data correspond to these days. For GCR-4 we approximate the Central Bank data linearly and then average over four days.
The dynamics of the Russian stock and bond market is subject to many factors both political and caused by the state of affairs in other sectors of the financial market, i.e., in the currency and state bond markets. It is worth noting that there is a high correlation between the stock market and gold and currency reserves of the Central Bank of the Russian Federation, as well as the stock markets in other countries. However, all these correlations are not stationary. Therefore, starting from political and economical factors that has been acting in Russia since 1995, we divide the observation time into stages. The passage from one stage to another is determined by certain events that happen outside the market but significantly affect the dynamics of its factors.
The natural bounds of these time intervals are important political and economical events that occurred in this period and significantly affect the participants in the Russian stock market. We divide the time-period from January 1995 to August 1998 into seven time stages by using the currency rate dynamics, the price dynamics for synthetic GKO bonds with 90-, 180-, 270-day payments periods, and the information on the RTS index variation which characterizes the Russian stock market outside the stock exchange.
Stage 1. January--May 1995
The first stage is the period in which the Russian State debt market was formed.
The currency corridor introduced at the end of 1994 and some other severe measures taken to stabilize the rouble gradually made the rates in GKO--OFZ market very attractive to investors (150-180% at the beginning of this stage), since they allowed high rates of currency return. In this period an important role is played by the active struggle with inflation. This struggle results in a significant decrease of this macroeconomics
factor (from 17.8% in January 1995 to 5.4% in July). At this stage the Government could evolve such rigid policy by using the stabilization credits given by the International Monetary Fund. In this period the dollar exchange rate grew from 3500 roubles per dollar till 5000 roubles per dollar.
Stage 2. June--July 1995
The second stage in the history of the Russian stock market is characterized by the final victory over inflation and a sharp decrease in the bounds of the currency corridor. It should be noted that during the summer months in 1995 the dollar exchange rate decreased till 4500 rouble per dollar (almost by 10% in contrast to May) and the inflation decreased till 2--3% per month.
Stage 3. August--December 1995
The third stage is completely determined by the bank crisis in August--September 1995 and the fact that investors were not confident in the results of the December elections to the State Parliament (Duma). To a large extent, the bank crisis was caused by low rates of interest in the State debt market, as well as by inability of several participants in the market to work when rates of interest are low and the currency market is stabilized. Some banks sold portfolios of GKO in order to maintain the current liquidity, which resulted in large oscillations in rates of interest in the State debt market.
In the second half of this period, the lack of confidence of investors in the results of the Parliament elections in 1995 was a destabilizing factor. It should be noted that the absence of big players in the stock market resulted in extremely low daily trade volumes.
It should also be noted that till the beginning of 1997 the scales of the stock market were significantly lower than the scales of the GKO--OFZ market. For example, with respect to such an important parameter as the daily trade volume, the stock market was more than 10 times smaller than the State debt market. It is not astonishing that the influence of the stock market on the economical life of the country at that time was insignificant.
Stage 4. January--June 1996
The fourth stage in the history of the Russian stock market is characterized by the pre-election fever in 1996. One can observe efforts to maintain the rouble exchange rate and to attract money resources necessary for successful per-election campaign. In April-- May the Central Bank of the Russian Federation sold more than 3 milliards dollars from its currency reserves. These problems were solved successfully (the rouble exchange rate varies only by 6% during the entire period, i.e., from 4700 roubles per dollar to 5000 roubles per dollar), which resulted in a sharp increase in rates of interest in the Russian stock market. To the end of this period, the GKO rates grew up to the annual interest of 250—300%. At this stage, without any doubt, one had to take into account the psychological factor that the majority of players were not sure in the further continuation of reforms. It should also be noted that the response of the stock market to the pre- election fever was different. Starting from March 1996, the plot of the RTS index shows a stable-increasing trend.
Stage 5. July 1996--October 1997
This stage is characterized by the President election, the end of the Chechen war, and the credit rating given to Russia by the International agencies. The further efforts to maintain the rouble exchange rate and to decrease inflation, as well as to attain political stabilization, resulted in a further decrease in rates of interest. In new issues one could see obvious trends to increase their periods of life; for instance, year-term GKO and OFZ with constant coupons appear. Russia received regular transfers of money from the International Monetary Fund in due time. In the second half of this period, another positive factor appeared: in November 1996 Russia obtained the credit rating BB+. To the spring of 1997 the rouble rate of annual interest of GKO attained 15--18%, which corresponds to 10--14% foreign exchange income. These rates of interest are comparable with rates of interest provided by national bonds of other countries with the same credit rating. All premises for further growth of prices and bond indices at this economical stage were exhausted. Before Russia received the credit rating, the price dynamics in the stock market was rather flaccid and the daily trade volumes rather low. After Russia received the credit rating, the share quotation in the stock market grew extremely fast till the end of the period. On October 6, 1997, the RTS index attained the record-breaking value of 571.66 points, the largest value during its entire history. At this stage, a relative increase in the RTS index was almost 500% during half a year. It should be noted that in this period the number of bonds passing the listing procedure in RTS and daily trade volumes grew tremendously. To the summer in 1997, the scales of the State debt market and the stock market became comparable. The dollar exchange rate varied weakly in the entire period and attained the level of 5900 roubles per dollar to the end of the stage. In fact, the end of this stage coincides with ‘’Black Tuesday'' 28 October, when the RTS index fell by 28%.
Stage 6. November 1997--March 1998
The world financial crisis attacked, to some extent, the markets in all countries over the world. In a relatively short period of time, the bond indices fell by 10--15%, while the RTS index dropped almost by one third. Attempts to stabilize the situation by maintaining the rouble exchange rate and increasing the rates of interest had only a local success and allowed a temporary stabilization in the State debt market at the beginning of March. To the beginning of March, the state of affairs in the stock market had also changed somewhat to the best, but the RTS index did not even approach the pre-crisis values. It should be noted that, in spite of the unfavorable situation in the foreign and Russian financial markets, the Government succeeded in maintaining the rouble exchange rate with respect to dollar and, on January 1, even denominated the rouble one thousand times. At the end of this stage, a political crisis struck Russia: the cabinet of Victor Chernomyrdin was forced to resign. At the same time the confrontation among the branches of the Government reinforced. In the middle of March the rating agencies decreased the credit rating of Russia. The prices in the world oil market began to fall violently, which decreased the income of the most of Russian oil companies and sharply decreased the returns obtained in foreign trade.
Stage 7. April--August 1998
The seventh stage is the period of the next collapse in the stock market. Efforts to maintain the rouble exchange rate and an increase in the rate of interest did not lead to a cardinal solution of the problems that appeared at the previous stage. The currency outflow from Russia got out of control and the pressure on the Central Bank grew violently. An actual threat of the rouble devaluation makes investors to sell their GKO portfolios in a hurry at dumping prices and to buy the currency for exportation. The positive feedback circuit is thereby closed. To the end of July the process got out of control of the Government completely: there were no new GKO trading, the payments for the old issues were covered directly from the budget, the rouble exchange rate was maintained by selling gold and currency reserves of the Central Bank. The oil crisis significantly restricted the possibilities of the Central Bank somehow to enlarge its gold and currency reserves by buying the currency from exporters. As a result, the Government defaulted on payments for GKO-OFZ during 1998--1999 years and declared a 90-day moratorium on foreign debts (this was an actual default); a large-scale rouble devaluation occurred. All this led to a system crisis in the entire country. Although the stock market continued to operate after August 17, the liquidity was lost to a large extent.
To the end of August the prices of the majority of bonds attained their minimum and the daily trade volume decreased by several tens of times.
Tables 2-3 show some general characteristics of the internal State debt market (Table 2) and the stock (Table 3) during the period 1995-1998 within the framework of the stages listed above.
Table 2.
Stage number
Stage bounds (duration in days)
Debt volume to the end of the stage (billion denom.
Roubles)
Variation in synthetic bond price
Synthetic bond rate of interest (averaged over the period) 1 January 1,1995-
May 31,1995 (150)
- 28.7% (for 90-
day bonds ) 58.9% (for 180- day bonds)
126.4% (for 90- day bonds 122.% (for 180- day bonds) 2 June 1, 1995-
July 31,1996 (61)
- -8.7% (for 90-
day bonds -6.5% (for 180- day bonds
55.8% (for 90- day bonds) 68.7% (for 180- day bonds)
3 August 1.1995- 76.7 0.4% (for 90- 94.0% (for 90-
December 17,1995 (139)
day bonds) -9.6% (for 180- day bonds)
day bonds) 114.0% (for 180- day bonds) 4 December 18,1995-
June 30, 1996 (195)
- 1.4% (for 90-
day bonds) 1.7% (for 180- day bonds)
92.3% (for 90- day bonds) 111.2% (for 180- day bonds) 5 July 1, 1996-
October 31,1997 (489)
- 17.9% (for 90-
day bonds) 37.5% (for 180- day bonds) 24.7% (for 270- day bonds)
30.8% (for 90- day bonds) 35.1% (for 180- day bonds) 25.9% (for 270- day bonds)
6 November 1,1997-
March 20, 1998 (140)
415.7 -0.9% (for 90-
day bonds) -3.2% (for 180- day bonds) -5.8% (for 270- day bonds)
21.6% (for 90- day bonds) 29.5% (for 180- day bonds) 30.5% (for 270- day bonds) 7 March 21, 1998-
August 14,1998 (146)
387.1 -24.6% (for 90- day bonds) -36.6% (for 180- day bonds) -44.7% (for 270- day bonds)
50.2% (for 90- day bonds) 53.5% (for 180- day bonds) 52.5% (for 270- day bonds)
Table 3.
Stage number
Stage bounds (duration in days)
Stock market capitalization at the beginning of the stage
RTS index variation.
Stage-averaged day trading volume in the RTS system
3 September 1,
995
18.0 mill. Doll. -17.6% 2.37 mill. Doll.
December 7,1995 (108)
4 December
8,1995 June 30, 1996
(195)
14.8 mill. Doll. 151.9% 8.77 mill. Doll.
5 July 1, 1996
October 31, 1997 (489)
37.2 mill. Doll. 106.8% 40.01 mill. Doll.
6 November
1,1997
March 20, 1998 (140)
76.6 mill. Doll. -20.5% 67.08 mill. Doll.
7 March 21, 1998
August 14, 1998 (146)
60.9 mill. Doll. -65.7% 45.83 mill. Doll.
On August 14, 1998 the stock market capitalization was 20.9 mill. Doll.
How can one explain the five-times increase in the price of the Russian stocks, while the general macroeconomics factors in Russia demonstrate at most only a tendency for a decrease in the rate of recession?
Even a cursory examination of Fig. 1 shows that there exists a high correlation between the dynamics of gold and currency reserves of the Central Bank and the dynamics of the RTS index in the period of its growth from the end of 1996 till its fall after October 27, 1997. The correlation between the gold and currency reserves of the Central Bank and the RTS index, calculated over the period from January 31, 1996, to August 31, 1998, was 0.74 and the correlation calculated over the period from December 31, 1996, to August 31, 1998, was even larger, 0.86.
The RTS index in this period increased approximately by a factor of five and than fell somewhat lower than its initial level. Therefore, we can assume that the volume of money resources attracted to the Russian stock market in this period was significantly larger than that attracted in 1995 and at the beginning of 1996. Unfortunately, we do not have a possibility to receive any information on this volume. Since the stock market participants themselves did not have such information, we can assume that the data on the gold and currency reserves of the Central Bank given in mass media can be a reference point common to everybody, especially under the conditions of comparatively slowly varying fundamental money factors (Figs. 2 and 3 in Appendix 1).
Generally speaking, in the period from 1995 to the present time (which is not a good time for the Russian stock market), one can observe the following two stable characteristic features. The first is that the majority of the market participants believe that the stocks of many Russian companies are underestimated. Different agencies and broker companies gave different estimates, but most of them confirm the fact that the prices of Russian stocks are significantly lower than the prices of western analogs. As is known, the common expectation of the market participants necessarily manifests itself in some way. The second characteristic feature of Russian stocks is that the dividends of ordinary stocks are extremely low if paid at all. To our knowledge, if for the stock price we take the stock price on September 1, 1998, and for the dividends we take the dividend magnitude in 1997, then the ratio D/P is approximately equal to 0.04 (this corresponds to the data given in the journal «Expert»). If we compare the 4-percent (with respect to the low price of the stock) annual dividends with the existing refunding rate approximately equal to the annual interest of 20--40%, then we can see that the purchase of stocks of Russian companies by nonstrategic investors is in character a game intended to increase the stock price. (The functioning of the futures market of the most liquid Russian stocks allowed additional speculative games.)
Just this second «peculiar» feature of the Russian stock market denudes the stock course price of some natural lower bound. The price can fall to a very low level. Besides of this, this fact practically annihilates any correlation between the stock course price and the actual situation in the corresponding industry branch.
However, here we must stress the following fact. If a company in the Russian stock market becomes attractive to foreign investors, then the market reacts to this fact immediately.
So, we can hypothesize that the Russian stock market is very attractive to foreign investors from the viewpoint of games intended to increase the stock price.
The infrastructure of the Russian stock market is well organized (more exactly, the infrastructure can readily be reconstructed if money resources are available) but ... so far the risks of operations in Russia are very high. If the risk decreases (provided that the problem of the debt reconstruction is solved, the foreign stabilization loans are given, and political stability is attained), then the stock market can regain its former liquidity in a very short period of time.
Although there are such striking peculiar features, the Russian stock market is not completely isolated, first of all, because of foreign investors.
As we can see in Fig. 1, these time-series have brightly expressed trends and, according to the information in Table 3, these trends are highly correlated in several pairs. It is natural to study the problem of cointegration of these time series. But a statistical analysis of time series of first differences of the variables under study (see Fig.
2) allows us to conclude that these series are nonstationary. Therefore, we cannot say that these series are cointegrated, at least, in the classical sense.
Is the time dependence between the stock index, the gold and currency reserves, and the state bond prices a special characteristic feature of the Russian financial market?
Studying the dynamics of stock markets in different countries during the last three years, we see that not only in the Russian market one could observe a five-times increase in the stock index followed by its drop to the original level. During 1996-98 the Venezuelan stock market varied in a similar way, and moreover, these changes were accompanied by variations in the gold and currency reserves just as it happened in Russian.
Y\ey_lkyebgZ[ex^Z_fZyaZ\bkbfhklvf_`^mnhg^h\ufbg^_dkhf
Table 4. Correlation matrix
RTS GKPR_180 GKPR_90 GCR_4
RTS 1.000000 0.802665 0.730466 0.733617
GKPR_180 0.802665 1.000000 0.972752 0.429397 GKPR_90 0.730466 0.972752 1.000000 0.407460 GCR_4 0.733617 0.429397 0.407460 1.000000
Fig. 1 Time dependence of the RTS, GKPR_180, GKPR_90, GCR_4 series
0 100 200 300 400 500 600
50 100 150 200 250 300 350 400 450 500
RTS
12 14 16 18 20 22 24 26
50 100 150 200 250 300 350 400 450 500
GCR_4
40 50 60 70 80 90 100
50 100 150 200 250 300 350 400 450 500
GKPR_180
65 70 75 80 85 90 95 100
50 100 150 200 250 300 350 400 450 500
GKPR_90
Fig. 2 Time dependence of differences
-120 -80 -40 0 40 80
50 100 150 200 250 300 350 400 450 500
DRTS
-0.6 -0.4 -0.2 0.0 0.2 0.4
50 100 150 200 250 300 350 400 450 500
DGCR_4
-10 -5
0 5 10 15
50 100 150 200 250 300 350 400 450 500
DGK_180
-10 -5
0 5 10 15
50 100 150 200 250 300 350 400 450 500
DGK_90
An analogy with the Russian stock market can be observed for the following time series of the Venezuelan market:
IBC-stock index, Reseves$ - international reserves in mill. Doll., ResevesBs - international reserves in bolivars, BP_30, BP_60, BP_90 - the prices of synthetic state bonds with nominal value of 100 and with 30-, 60-, 90-day periods of payment. The time dependence of the series IBC and Reserves$, BP_30 and BP_60 and BP_90, DReserves$
and DReservesBs during 2.5 years from January 2, 1997, to August 5, 1999, is shown in Fig. 3. One can readily see that these series have distinct trends and the correlation between IBC and ReservesBs is very large. If we neglect details, then we see that the dynamics of time series has much in common for Russia and Venezuela (Figs. 1 and 3).
Fig. 3 Dynamics of the finanacial market in Venezuela
12000 14000 16000 18000 20000
100 200 300 400 500 600 Reserves$
2000 4000 6000 8000 10000 12000
100 200 300 400 500 600 IBC
86 88 90 92 94 96 98
100 200 300 400 500 600
BP_90
000000 500000 000000 500000 000000 500000
100 200 300 400 500 600 ReservesBs
88 90 92 94 96 98 100
100 200 300 400 500 600 BP_60
95 96 97 98 99 100
100 200 300 400 500 600 BP_30
2. Arbitrage and quasi-arbitrage operations in the cash and futures markets Futures contracts in the stock market
Securities are financial tools based on real assets. Financial derivatives are financial tools based on other simpler financial tools. It is commonly agreed to classify forward and futures contracts, options, financial futures and financial futures options as financial derivatives. In turn, financial futures as divided into currency futures, interest futures (for promissory notes), and swaps and futures for stock market indices.
When the financial derivatives are bought, they impose certain obligations on the seller and the buyer concerning a fixed time instant in the future (or until a fixed time instant in the future) and the tools that underlie the financial derivatives. Transactions of such type are called forward transactions (transactions for a term) in contrast to transactions in the cash market (cash or spot transactions) with immediate payment and delivery. A classical example of spot transactions is buying (selling) of a forward contracts.
A forward contract is an agreement between two parties one of which commits oneself to buy and the other party commits oneself to sell a certain amount of goods or securities at a specified price at a future time instant fixed in the agreement (on the day at which the contract is executed). In this case neither money nor goods are required till the time indicated in the contract. Since there are no standards for terms and amounts of goods, forward contracts are used only out of stock markets.
{Strictly speaking, according to this definition, goods forwards and futures cannot be classified as financial derivatives, since they are based on actual goods but not on financial tools. However, in the literature, this specific characteristic of goods forwards (futures) is not taken into account when possible combinations of financial derivatives and real assets are analyzed.}
Futures contracts are traded in stock markets. A difference between futures and forward contracts is that the delivery amount of certain goods is fixed in the futures contract (both in nomenclature and in quality) and futures contracts must be executed in certain fixed months of the year. In this case, the stock market guarantees that the contract will be executed and, in this sense, it is absolutely unimportant for the buyer (seller) who, in fact, executes the contract (i.e., who delivers the goods or securities and who pays for the delivery). In the stock market jargon, to buy a futures contract means to take a long position (to go long) and to sell a futures contract means to take a short position (to go short). If a participant of the futures market simultaneously goes long and short in the same futures contract, it is said that the position of this participant in this futures contract is closed.
To fulfill the obligations assumed, the stock market obliges both the seller and the buyer of futures to deposit some money resources (the initial margin) in the Settlement House of the market. This initial margin will be returned when the contract is executed or the position is closed. The quantity of the initial margin depends on the concrete assets and is determined by the market with account of the volatility of the price of the assets.
In the Russian market the quantity of the initial margin is usually equal to 5% of the price of the contract. In some markets it is allowed to deposit up to 70% of the initial margin in liquid securities indicated in advance.
Besides of the initial margin deposited before a position is taken, the market obliges all participants with open positions to deposit some additional money to compensate the negative (from the viewpoint of those in open positions) daily variation of the price of the contract, i.e., the so-called negative variation margin. For this purpose, every day the
calculated (settlement) price is announced in the market. This price is calculated over the results of the daily auction (often, it is simply the closure price, i.e., the price during the last minutes of the auction). Then the daily variation in the calculated price (the settlement change) is announced as the difference between the calculated prices of the current and preceding days. On the basis of these data, the Settlement House of the market reestimates positions of the participants, i.e., performs marking them to the market.
If, in this case, it turns out that the settlement change of the contract is positive, there are the following consequences:
• the variation margin is positive for those who go long in this contract (the sum equal to the settlement change will be paid into his current account);
• the variation margin is negative for those who go short in this contract (the sum equal to the settlement change will be taken from his current account). If the settlement change of the contract turns out to be negative, then the situation is opposite: those who go long «lose», while those who go short «win». Thus the absolute value of the variation margin is equal to the absolute value of the variation in the current account corresponding to the open position, while these variations are of opposite signs for the long and short positions.
The money in the current accounts of the contract buyers and sellers in the Settlement House in the form of the initial margin provide the possibility for the market to cover losses of both parties on any day if the prices volatility is unfavorable (the variation margin is negative). At the end of each trading day (before the next trading day starts), both the buyer and the seller must guarantee that their current accounts in the Settlement House contain money not less than a certain sum. This sum is called the maintenance margin. As, because of a negative daily variation margin, the sum of money in the current account decreases till the value of the maintenance margin (or becomes less than this value), the seller (buyer) of the contract must add to his current account the amount of the initial margin. If this requirement is violated, the Exchange on its own may close this unprovided position at the beginning of the next trading day.
Basic parameters that provide a gain in arbitrage transactions
Arbitrage transaction is a set of operations in the market that bring an income because of the difference in prices existing at the moment in one or several markets.
Arbitrage transactions without use of the ownership initial capital are often called pure arbitrage. Under certain conditions an arbitrage transaction can turn into pure arbitrage.
As an example, we consider the situation in the spot market of securities and in the market of borrowed capital. If only the tools of these markets are used, then the income obtained by sale and purchase operations depends on the future price of the assets. Since arbitrators do not know the future dynamics of prices, one can speak about each arbitrage transaction with some probability (if we mean the future) or as of a lost possibility (if we mean the past). Hence we can treat each transaction not as an arbitrage but as a speculative transaction.
A more definite situation arises if there exists both cash and forward (futures) markets for a certain type of goods. To analyze a typical situation in the cash, futures, and money markets, we need the following notation. For the unit of goods we take 100 stocks of a company whose dealer sale and purchase prices at the initial time t are equal to Pa(t) and Pb(t), respectively, (Pa(t)>Pb(t)) and whose corresponding prices of execution of futures contracts (at time T ) are
FaT(t) (for the buyer of the contract) and FbT(t) (for the seller of the contract) (FaT(t) ≥ FbT(t)).
We assume that the annual borrowing rate possible for the arbitrator is rbor(t) and the possible lending rate is rlen(t). Moreover, we assume that
a) the transaction costs for buying 100 stocks of a company are equal to the fraction rtr of the purchase price;
b) the loan of 100 stocks of a company for the term (T-t) is equal to the fraction ρloan of the sale price at the time of signing the loan contract;
c) the initial margin in the futures market is equal to the fraction rm of the contract price.
By using the above notation, one can easily analyze the payment flows arising in arbitrage transactions of two types (the direct and reverse cash-and-carry strategies).
Description of the direct arbitrage operation (cash-and-carry strategy) At the current time t
1) To take the credit B=Pa(t)*(1+rtr)+FbT(t)* rm for the period T-t at the 100*rbor(t) annual rate.
2) To buy 100 stock of a company in the cash market at the price Pa(t) and to pay the transaction costs Pa(t)* rtr..
3) To sell the futures contract (going short) for delivery of 100 stocks of the company after the time interval T-t at the price FbT(t) and to deposit the initial margin FbT(t)* rm
At time T of the contract execution
1) To deliver 100 stocks of the company according to the futures contract. To receive, according to the contract, the sum of money FbT(t) and the initial margin FbT(t)*
rm.
2) To return the sum of the money loan which together with the interest is [Pa(t)*(1+rtr)+ FbT(t)* rm ]*(1+rbor(t)*(T-t)).
The gain received during the time interval T-t is equal to
)]
(
* ) (
* )
(
* ) ( 1 [
* ) (
)) (
* ) (
* 1 ( ) (
)) (
* ) ( 1 (
* ]
* ) ( ) 1 (
* ) ( [ ) 1 ( )
1 (
t T t r r r t T t r t
P
t T t r r t
F
t T t r r
t F r t
P r t
F
bor tr tr bor
a
bor m b
T
bor m
b T tr a
m b
T
− +
+
− +
−
−
−
∗
=
=
− +
+ +
− +
∗
= Π
(2.1)
The conditionΠ1>0 can be written in the form
) , ( )
,
(t T t UBpa t T t
ra − > −
(2.2) where the direct arbitrage interest rate ra(t,T −t) is defined by the relation
t T t
P t P t t F
T t
r a
a b
T
a −
= −
− 1
) * (
) ( ) ) (
,
( , (2.3) and the symbol UBpa(t,T −t) stands for the Upper Bound for pure arbitrage
] )* (
) (
* 1 ) 1 (
[
* ) ( ) ,
( a m
b T bor
tr tr
bor r
t P
t F t T t r r r t
r t T t
UBpa +
+ − +
=
− (2.4)
Description of the reverse arbitrage operation (reverse cash-and-carry strategy) At the current time t
1) To loan 100 stocks of a company for the period T-t.
2) To sell 100 stocks of the company in the cash market at the price Pb(t) and to pay the transaction costs Pb(t)* rtr.
3) To buy a futures contract (to go long) for delivery of 100 stocks of the company after the time interval T-t at the price FaT(t) and to deposit the initial margin FaT(t)* rm in the Settlement House of the market.
4) To deposit the remaining sum of money L(t)= Pb(t)*(1-rtr)- FaT(t)* rm for the time interval T-t at the rate of 100* rlen(t) annual return.
{It is assumed that L(t)>0, which, in turn, bounds the value FaT(t):
FaT(t)<Pb(t)*(1-rtr)/rm. Since the variables rm and rtr are small (rm<0.1 and rtr<0.1), this inequality holds practically always.}
At time T of the contract execution
1) To receive the sum [Pb(t)*(1-rtr) - FaT(t)* rm](1+rlen(t)*(T-t)) of money deposited.
2) To receive the delivered 100 stocks of the company according to the futures contract and, according to the contract, to pay the sum of money FbT(t) and to receive the initial margin FbT(t)* rm.
3) To return 100 stocks and to pay the sum Pb(t)* ρloan for the money loan.
{In transaction 2, the combination of operations 1) and 2) at time t and operation 3) at time T is called the short sale or the sale without settlement. This concerns any possible assets, not only god. However, such an operation is not always possible because of various reasons.}
The gain received during the time interval T-t is equal to
)) (
* ) (
* 1 (
* ) (
)]
(
* ) (
* )
(
* ) ( 1 [
* ) (
* ) ( ) (
* ) ( )) (
* ) ( 1 (
* )
2 (
t T t r r t
F
t T t r r r
t T t r t
P
t P t F r t F t T t r t
L
len m a
T
len tr loan tr len
b
loan b
a T m a T len
− +
−
−
−
−
−
− +
=
=
−
− +
− +
= Π
ρ
ρ
(2.5)
The condition Π2 >0 can be written in the form
] )* (
) (
* 1 ) (
* 1 ) 1 (
[
* ) ( ) , (
), , 1 (
) * (
) ( ) ) (
, (
b m a T len
loan len
tr tr len
b b a
T a
t r P
t F t T t r t T t r r r t
r t T t LBpa
t T t t LBpa T
t P
t P t t F
T t r
− −
− −
−
−
=
−
−
− <
= −
′ −
ρ (2.6)
Here, by analogy with (2.3) and (2.4), we use the following notation: ra′(t,T −t) stands for the reverse arbitrage interest rate and LBpa(t,T−t) for the Lower Bound of pure arbitrage.
Let us note the following characteristic features of arbitrage strategies:
1. The cash-and-carry strategies considered do not use the initial ownership capital.
Hence the usual notion of gain cannot be used for characterizing such cash-and-carry strategies. However, the norms of income of the direct and reverse arbitrage operations defined by relations (2.3) and (2.6) characterize the relation between the futures and spot markets under existing loan and deposit rates.
2. It is unprofitable to carry out the direct and reverse cash-and-carry strategies simultaneously, since we have
2 0
1+Π <
Π ,
in view of the inequalities FaT(t)-FbT(t)>0, Pa(t)-Pb(t)>0, rbor-rlen>0.
3. The assumption that it is impossible to realized a pure arbitrage transaction implies the following natural two-sided bounds on the prices of futures contracts:
)) (
* ) ( 1 (
* ) ( ) ( ) ( )) (
* ) ( 1 (
* )
(t T t LBpa t F t F t P t T t UBpa t
Pb + − < Ta ≈ Tb < a + − (2.7)
4. If we neglect the spread between the purchase and sell prices, then we have )
, ( ) ,
(t T t r t T t
ra′ − ≈ a − . In this case the two-sided inequality (2.7) turns into the following two-sided inequality for the norm of income of the arbitrage operation: the direct and reverse cash-and-carry strategies of pure arbitrage carried out at time t during the period T-t are unprofitable if
) , ( )
, ( ) ,
(t T t r t T t UBpa t T t
LBpa − < a − < − (2.8)
In other words, a situation, which theoretically is arbitrageless, can exist only if inequality (2.8) is satisfied.
Synthetic bond and quasi-arbitrage
In the preceding section we studied the situation in which it was possible to realize the pure arbitrage. However, if the ownership capital exists, a transaction of cash-and- carry type can be regarded as an alternative to riskless investments in Treasury bills.
Under certain conditions, this alternative transaction is more profitable than investment in the state security bonds. A realization of this possibility is called quasi-arbitrage.
Let us study this possibility in more detail. We assume that an investor does not want to loan money and is ready to spend the sum B=Pa(t)* (1+rtr)+FbT(t)* rm of his own money in order to realize a direct arbitrage transaction. Then the following two operations are excluded from the cash-and-carry strategy: one need not loan money and return this money with interest .
We see that, for the operation under study, the payment flow is equivalent to the payment flow in the case of purchasing a discount couponless bond and possessing this bond till its repayment. Such a bond, which does not exist in nature, is called a synthetic bond. In other words, a synthetic bond is a set of operations with financial tools whose payment flow is equivalent to the payment flow of a couponless bond. One can define a synthetic bond by the following symbolic relation:
Synthetic T-bill = Spot-Futures
For the initial costs of size B(t), the profit obtained by the investor is equal to ]
1 [
* ) ( )
1 ( tr
a b
T t P t r
F − +
=
Π ,
and the income of this transaction (in other words, the lending rate on the synthetic bond market calculated by the simplest interest formula with one-year time base) is equal to
) (
* ]
* ) ( ) 1 (
* ) ( [
) 1 (
* ) ( ) ( )
(
* ) ) (
,
~( 1
t T r t F r t
P
r t
P t F t
T t t B T t r
m b T tr a
tr a
b T
− +
+
+
= −
−
= Π
− . (2.9)
Suppose that, at the same time, the lending rate on the GKO market with the payment period T-t is equal to r~ (t,T-t). Then, by definition, the conditionlen
~r(t,T−t)>~rlen(t,T −t) (2.10) is a condition under which the quasi-arbitrage is possible for this investor. In this case, the price of the futures contract FbT(t) must satisfy the inequality
)]
(
* )
~ (
* ) (
* )
~ ( 1
[
* ) ( )) (
~ *
* 1 (
* )
(t r r T t P t r r t T t r r t T t
FTb − m len − > a + tr + len − + tr len − .
By analogy with UBpa(t,T −t), we define the upper bound of the quasi-arbitrage operation as follows:
] )* (
) (
* 1 ) ,
~ ( 1
[
* ) ,
~ ( ) ,
( a m
b T len
tr tr
len r
t P
t F t T t T t r r r t
T t r t T t
Ubqa +
− + −
+
−
=
− . (2.11)
Then we can easily rewrite (2.9) as
) ) (
, (
) ) (
, (
* ) ,
~ ( ) ,
~(
t T t r
T t Ubqa
t T t r
T t r t T t r t T t r
tr tr a
len
− −
−
− −
− −
=
− . (2.12)
Hence the quasi-arbitrage realization condition (2.10) acquires the form
ra(t,T−t)>Ubqa(t,T−t). (2.13) If we compare the opposite inequality, which means that the quasi-arbitrage is impossible, with the inequality Π1 <0, which means that the pure arbitrage is impossible, then we can easily see that the only difference between them is that rbor is replaced by r~ . If len rbor >~rlen , then Ubqa(t,T−t)>UBpa(t,T−t). However, for a more realistic
condition ~ 1
1
0< − <<
bor len
r
r , the following inequality is satisfied:
) , (t T t
Ubqa − <UBpa(t,T −t), i.e., the condition that the pure arbitrage is impossible is more rigid than that for the quasi-arbitrage. This means that even if the pure arbitrage is impossible in the market (for investors who do not have their own capital), a realization of quasi-arbitrage is possible for investors who have their own initial capital. Recall that the initial finance state (the bench mark) of an investor realizing the quasi-arbitrage is different than that of an investor realizing the pure arbitrage.
Reverse quasi-arbitrage
Assume that an investor possesses 100 stocks of a company (there exist futures contracts for these stocks) and a sum of money B(t)=FTa(t)*rm +Pb(t)*rtr. He sells 100 stocks at the price Pb(t), pays the transaction costs Pb(t)*rtr, and buys a futures contract for 100 stocks of this company with payment period T . For this futures contract he pays the initial margin FTa(t)*rm. Then the investor deposits the money obtained for the stocks at the rate of 100* rlen(t) annual return and thus receives the sum Pb(t)*(1+rlen(t)*(T-t)) on the day of the contract execution. He pays the price of the contract )FTa(t from the money obtained and gets the 100 stocks back.
As the result of these operations, the investor has the same 100 stocks plus the money profit
tr b a
T len
b t r t T t F t P t r
P ( )*(1 ( )*( )) ( ) ( )*
2 = + − − −
Π (2.14)
The rate of return of the above operation can be calculated as the ratio of the profit (2.14) to the cash expenditures divided by the time interval between the beginning and the end of the operation:
) (
* ]
* ) (
* ) ( [
) ( ] ) (
* ) ( 1 [
* ) ( ) (
* ) ) (
,
ˆ( 2
t T r t P r t F
t F r t T t r t
P t t t t B T t r
tr b m a T
a T tr len
b
− +
−
−
−
= +
−
= Π
− (2.15)
By analogy with (2.10), the following inequality is a condition that a quasi-arbitrage realization does not lead to losses for the investor:
) ,
~ ( ) ,
ˆ(t T t r t T t
r − > len − . (2.16)
By using notation (2.6) and taking into account (2.15), we can rewrite condition (2.16) in the form
) )* (
) ( (
* )]
( ) ,
~ ( [
] )* (
) ( ) (
* ) 1 (
[
* ) ( ) , (
), , ( )
, (
b m a T tr ltn
len
b m a T len
tr tr
len a
t r P
t r F
t r t T t r
t r P
t F t T t r r r t
r t T t Lbqa
t T t Lbqa t
T t r
+
−
−
−
− −
−
−
=
−
−
<
′ −
(2.17)
If the spread between the purchase and sell prices is neglected, then we have )
, ( ) ,
(t T t r t T t
ra′ − ≈ a − . In this case the condition that both the direct and the reverse quasi-arbitrage is impossible can be written in terms of a two-sided inequality for the norm of income of the arbitrage operation as follows:
) , ( )
, ( ) ,
(t T t r t T t Ubqa t T t
Lbqa − < a − < − (2.18)
For some natural relations between the existing rates of return, inequality (2.18) implies inequality (2.8). But inequality (2.8) does not necessarily implies (2.18).
Norm of income of arbitrage operations as an indicator of arbitrage possibilities
It follows from the above theoretical arguments that one can estimate the possibility of an arbitrage strategy by comparing the norm of income of the arbitrage operation
) , (t T t
ra − with the bounds UBpa(t,T−t), )Ubqa(t,T−t , Lbqa(t,T −t), )
, (t T t
LBpa − . Let us consider the case most frequently encountered in practice, i.e., the case of embedded inequalities
) , (t T t
LBpa − <Lbqa(t,T −t)<Ubqa(t,T−t)<UBpa(t,T −t)
for each t and T-t. All possible arbitrage realizations are summarized in Table Z, which is rather simple.
Table Z
Relation Profitable strategy
UBpa
ra > Both the pure direct arbitrage and the direct quasi-
